Question

If $\left|\begin{array}{ccc}p& q-y& r-z\\ p-x& q& r-z\\ p-x& q-y& r\end{array}\right|=0$, then the value of $\frac{p}{x}+\frac{q}{y}+\frac{r}{z}$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4pqrs$

Answer 1

The correct option is C $2$

Dividing $C_1,C_2,C_3$ by $x,y,z$ we have
$△=xyz\left|\begin{array}{ccc}\frac{p}{x}& \frac{q}{y}-1& \frac{r}{z}-1\\ \frac{p}{x}-1& \frac{q}{y}& \frac{r}{z}-1\\ \frac{p}{x}-1& \frac{q}{y}-1& \frac{r}{z}\end{array}\right|=0$
Applying $C_1\to C_1+C_2+C_3$
$\Rightarrow △=xyz\left|\begin{array}{ccc}\sum \frac{p}{x}-2& \frac{q}{y}-1& \frac{r}{z}-1\\ \sum \frac{p}{x}-2& \frac{q}{y}& \frac{r}{z}-1\\ \sum \frac{p}{x}-2& \frac{q}{y}-1& \frac{r}{z}\end{array}\right|=0$
$\Rightarrow xyz(\sum \frac{p}{x}-2)\left|\begin{array}{ccc}1& \frac{q}{y}-1& \frac{r}{z}-1\\ 1& \frac{q}{y}& \frac{r}{z}-1\\ 1& \frac{q}{y}-1& \frac{r}{z}\end{array}\right|=0$
Using $R_1\to R_1-R_2,R_2\to R_2-R_3$ then expanding with the help of third Row $\left(R_3\right)$ we get $\sum \frac{p}{x}=2$
i.e. $\frac{p}{x}+\frac{q}{y}+\frac{r}{z}=2$
Hence choice (c) is correct.